The $n$-valent convexity of Frasin integral operators

Abstract

UDC 517.5

Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form
$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$
We consider the Frasin integral operator as follows:
\begin{gather*}\label{e1.3}
G_{n}(z)=\int\limits_{0}^{z} n\xi^{(n-1)}\bigg(\dfrac{f'_{1}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{1}}\cdots\bigg(\dfrac{f'_{k}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{k}}d\xi.
\end{gather*}
In this paper, we obtain a sufficient condition under which this integral operator is $n$-valent convex and get other interesting results.